Our next goal is to construct the effective theory describing the dynamics and interactions of macroscopic electromagnetic fields in the quantum vacuum. Here, we focus on the limit of \(\lbrace E/E_{\rm cr},B/B_{\rm cr}\rbrace \ll 1\)  relevant for laboratory experiments; cf. the recent reviews [1]}, [2]}, [3]}, [4]} and references therein.
Correspondingly, a perturbative expansion of the vacuum-fluctuation-mediated effective interactions of the applied electromagnetic fields in powers of \(E\)  and \(B\) , or – resorting to a covariant and gauge invariant formulation – in powers of the field strength tensor \(F^{\mu \nu }\)  and its dual \({}^*\!F^{\mu \nu } =\frac{1}{2}\epsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }\) , can be truncated at low orders; \(\epsilon ^{\mu \nu \rho \sigma }\)  is the Levi-Civita symbol.
